Which of the following statements about public key cryptosystem is NOT true?

The statement regarding encoding a message solely with the private key and decoding it with the public key is not true within the context of public key cryptosystems. In such systems, messages can indeed be encoded using either key. Typically, a message that is intended for confidentiality is encrypted with the recipient's public key, which ensures only the holder of the corresponding private key can decrypt it.

On the other hand, using the private key to create a digital signature allows anyone to verify the signature using the public key, promoting integrity and authentication. Thus, while the private key does play a crucial role in the signing process and establishing authenticity, it is not the only method for encryption in a public key infrastructure; encoding can also occur with the public key.

The other statements accurately reflect the principles of public key cryptosystems. For instance, message encoding and decoding with complementary keys is a fundamental feature, and knowing the public key indeed does not compromise the private key's security. Additionally, such systems are versatile in their applications, extending beyond mere encryption to include authentication and the creation of digital signatures, further emphasizing their robustness and utility.